Our goal is to find the supremum cosine angle between multiplication invariant (MI) spaces using range functions and study the conditions under which the two MI spaces will be closed. The MI spaces were first introduced by M. Bownik and K. Ross [1]}. The study of such spaces is associated with the study of the Bessel system generated by multiplications in \(\mathcal {H}\) -valued Bochner space \(L^2(X;\mathcal {H}).\)  The idea is to generalize the modulation invariant subspaces of LCA groups and the same is done by introducing the concept of Parseval determining set in \(L^1(X),\)  which is the generalization of characters on an LCA group. The main advantage is that the results developed for the supremum cosine angle between MI spaces will be utilized to find the supremum cosine angle between translation invariant subspaces of locally compact groups using Zak transform which converts the translation into multiplications [2]}. This setup covers all the classical ones like shift-invariant systems of \(L^2(\mathbb {R}^d)\) . The main reason to choose the MI spaces is that not only we can develop similar results for shift invariant spaces but also for translation invariant spaces where the translations are taken over a non-discrete subgroup of LCA group using the Zak transformation. In this regard, it generalizes the classical results related to the closedness of two \(\Gamma \) -translation invariant subspaces in the context of a locally compact group \(G\)  (need not be abelian), having \(\Gamma \)  as a closed abelian subgroup. One of the benefits of Zak transform for the pair \((G, \Gamma )\)  is that the various inaccessible examples pairs like \((\mathbb {R}^n, \mathbb {R}^m)\) , \((\mathbb {R}^n, \mathbb {Z}^m)\) , \((\mathbb {Q}_p, \mathbb {Z}_p)\) , \((\mathcal {G}, \Gamma )\) , etc., can be accessed through it where \(n\ge m\) , \(\Gamma \)  (not necessarily co-compact, i.e., \(\mathcal {G}/\Gamma \) -compact, or uniform lattice) is a closed subgroup of the second countable locally compact abelian (LCA) group \(\mathcal {G}\) , and \(\mathbb {Z}_p\)  is the \(p\) -adic integer in the \(p\) -adic number \(\mathbb {Q}_p\) . In case of \(\Gamma \)  co-compact, we have a similar characterization using fiberization. First, we characterize the supremum cosine angle using Gramian and dual Gramian operator (Theorem REF ). The Gramian and the dual Gramian analysis is more appropriate for the analysis of a Riesz basis and frame respectively. The Gramian analysis is more helpful for the case of finitely many generators as then the MI spaces can be characterized nicely using a finite matrix instead of an infinite matrix for the case of infinitely many generators.
